# Difference between revisions of "Lower attic"

From Cantor's Attic

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* [[Aleph#Aleph_one| $\omega_1$]], the first uncountable ordinal, and the other uncountable cardinals of the [[middle attic]] | * [[Aleph#Aleph_one| $\omega_1$]], the first uncountable ordinal, and the other uncountable cardinals of the [[middle attic]] | ||

* [[stable]] ordinals | * [[stable]] ordinals | ||

− | * [[Heights of models | + | * [[Heights of models]] <!--(ZFC without powerset axiom) is much above $\Sigma_n$-admissible, much below ZFC (stable ordinals as part of ZFC have no consistency strength)--> |

* The ordinals of [[infinite time Turing machines]], including | * The ordinals of [[infinite time Turing machines]], including | ||

** [[infinite time Turing machines#Sigma | $\Sigma$]] = the supremum of the accidentally writable ordinals, | ** [[infinite time Turing machines#Sigma | $\Sigma$]] = the supremum of the accidentally writable ordinals, |

## Revision as of 17:22, 22 September 2021

Welcome to the lower attic, where the countably infinite ordinals climb ever higher, one upon another, in an eternal self-similar reflecting ascent.

- $\omega_1$, the first uncountable ordinal, and the other uncountable cardinals of the middle attic
- stable ordinals
- Heights of models
- The ordinals of infinite time Turing machines, including
- bad ordinals
- reflecting ordinals
- admissible ordinals and relativized Church-Kleene $\omega_1^x$
- Church-Kleene $\omega_1^{ck}$, the supremum of the computable ordinals
- the omega one of chess
- $\omega_1^{\mathfrak{Ch}_{\!\!\!\!\sim}}$ = the supremum of the game values for white of all positions in infinite chess
- $\omega_1^{\mathfrak{Ch},c}$ = the supremum of the game values for white of the computable positions in infinite chess
- $\omega_1^{\mathfrak{Ch}}$ = the supremum of the game values for white of the finite positions in infinite chess

- the Takeuti-Feferman-Buchholz ordinal
- the Bachmann-Howard ordinal
- the large Veblen ordinal
- the small Veblen ordinal
- the Extended Veblen function
- the Feferman-Schütte ordinal $\Gamma_0$
- $\epsilon_0$ and the hierarchy of $\epsilon_\alpha$ numbers
- indecomposable ordinal
- the small countable ordinals, such as $\omega,\omega+1,\ldots,\omega\cdot 2,\ldots,\omega^2,\ldots,\omega^\omega,\ldots,\omega^{\omega^\omega},\ldots$ up to $\epsilon_0$
- Hilbert's hotel and other toys in the playroom
- $\omega$, the smallest infinity
- down to the parlour, where large finite numbers dream